Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = C23×AGL(1,𝔽7) (order 336) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #1131⋊C3 |
| Fitting | C7×C8 ⋊ C2×C3 |
| Hall | #1131 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ M16 |
| Sylow | C7 ⋊ C3×M16 |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,7)(3,4,6)(8,9,10,13,11,12,14,15);(2,3)(4,6)(5,7)(9,12)(13,15)Polycyclic presentation:
2^2,3,2^2,7|b^2=d,d^2=e,b^a=be,f^a=f^6,f^c=f^438 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1131⋊C3, #6464⋊C3, M16×AGL(1,𝔽7), … |
| maximal normal subgroup of | C2×#1131⋊C3, #6464⋊C3, M16×AGL(1,𝔽7), … |
| maximal quotient of | (ΓL(1,𝔽8)⋊C4)⋊C8, #6389⋊C3, (C2×AGL(1,𝔽7))⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 8);